Question

4. Let B be a matrix such that -2a -3b nullspace(B)- What is the dimension of the column space of B? What is the dimension of the row space of B?

Answer 1

here dimension of null space of B=2 and B is a m×6 matrix.hence dimension of columns space B=6-2=4 by rank nullity theorem. But we know that dim(column space B)=dim(row space B).

Given nullspace of B = {[2a + b -2a -3b a + b a b]} therefore dimension of nullspace of B = 2 Since basis for nullspace has only two variable, a & b Let X = [2a + b -2a -3b a + b a b] = nullspace (B). therefore BX = 0 Let B be a m times n Matrix here X is a 6 times 1 matrix. matrix multiplication is only possible if n = 6. Therefore 6 = dim(nullspace B) + Rank(B) = 2 + Rank(B) Rank(B) = 6 - 2 = 4 therefore dim(column space B) = Rank B = 4 Since dim(row space B) = dim(column space B) = 4 here dimension of null space of B = 2 and B is a mtimes 6 matrix. hence dimension of columns space B = 6-2 = 4 by rank nullity theorem. But we know that dim(column space B) = dim(row space B).